To solve the problem, we aim to minimize the function $L(x) = \frac{7}{\cos(x)} + \frac{2}{\sin(x)}$, where $x$ is the angle the ladder makes with the ground. We'll calculate the critical points by taking the derivative of $L(x)$ with respect to $x$ and solving for when the derivative equals zero.

Steps:

1. Derivative of $L(x)$: Using the chain rule and the derivatives of trigonometric functions: $L'(x) = -\frac{7 \sin(x)}{\cos^2(x)} + \frac{-2 \cos(x)}{\sin^2(x)}.$

   Simplify: $L'(x) = -\frac{7 \sin(x)}{\cos^2(x)} - \frac{2 \cos(x)}{\sin^2(x)}.$

2. Set $L'(x) = 0$: $-\frac{7 \sin(x)}{\cos^2(x)} = \frac{2 \cos(x)}{\sin^2(x)}.$

   Cross-multiply to eliminate fractions: $-7 \sin^3(x) = 2 \cos^3(x).$

   Rearrange: $\frac{\sin^3(x)}{\cos^3(x)} = -\frac{2}{7}.$

   This can be rewritten as: $\tan^3(x) = -\frac{2}{7}.$

3. Solve for $\tan(x)$: $\tan(x) = \left(-\frac{2}{7}\right)^{1/3}.$

   Compute the approximate value of $\tan(x)$ and use it to find $x$.

4. Evaluate $L(x)$: Substitute the critical value of $x$ back into $L(x)$ to find the length of the shortest ladder.

I will compute this numerically.import numpy as np

Define the function L(x) and its derivative

def L(x): return 7 / np.cos(x) + 2 / np.sin(x)

def dL(x): return -7 * np.sin(x) / (np.cos(x) ** 2) - 2 * np.cos(x) / (np.sin(x) ** 2)

Find the critical point numerically

from scipy.optimize import minimize_scalar

Minimize L(x) for x in the range (0, pi/2) since the angle x must be acute

result = minimize_scalar(L, bounds=(0, np.pi/2), method='bounded')

The angle and the shortest ladder length

x_min = result.x # angle that minimizes L(x) L_min = result.fun # minimum ladder length

x_min, L_min(0.582420798994774, 12.017944340100026)(0.582420798994774, 12.017944340100026)successThe shortest ladder length that clears the fence and reaches the building is approximately 12.02 feet. This occurs when the angle $x$ between the ladder and the ground is approximately 0.582 radians (or about 33.36°).

Key Steps:

1. The function $L(x) = \frac{7}{\cos(x)} + \frac{2}{\sin(x)}$ was minimized.
2. The optimal angle $x$ was found numerically, giving the shortest ladder length.

Let me know if you'd like further clarifications, or I can provide details of the computation process.

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Related Questions:

1. How does the placement of the fence affect the length of the ladder?
2. Can you derive the formula for $L'(x)$ step-by-step?
3. What would happen to the ladder length if the fence height increased to 3 feet?
4. How does the angle of the ladder influence the total length practically?
5. Can this problem be solved using geometric methods instead of calculus?

Tip: Always check if the constraints of a real-world problem match the calculated solution to ensure practical feasibility.